Question: All of the 3rd grade teachers and students from Santa Rita went on a field trip to an archaeology museum. Tickets were $$7.50$ each for teachers and $$3.50$ each for students, and the group paid $$32.50$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$30.00$ each for teachers and $$8.00$ each for students, and the group paid $$100.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+3.5y = 32.5}$ ${30x+8y = 100}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-30x-14y = -130}$ ${30x+8y = 100}$ Add the top and bottom equations together. $ -6y = -30 $ $ y = \dfrac{-30}{-6}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {7.5x+3.5y = 32.5}$ to find $x$ ${7.5x + 3.5}{(5)}{= 32.5}$ $7.5x+17.5 = 32.5$ $7.5x = 15$ $x = \dfrac{15}{7.5}$ ${x = 2}$ You can also plug ${y = 5}$ into $ {30x+8y = 100}$ and get the same answer for $x$ ${30x + 8}{(5)}{= 100}$ ${x = 2}$ There were $2$ teachers and $5$ students on the field trips.